Rule: Q[8, 13, 84]

Type

This rule is of Hermite (probabilists') type.

Polynomials

One starts from the polynomial P1 and extends one by one with Pi until the final polynomial defining this rule is reached.
Note: Polynomials might be truncated due to size! The complete polynomials are included in the download.

Node locations in the complex plane

Plot of the node locations in the complex plane.

Magnitude of the Weights

Logarithmic magnitude of the weights
         plotted against the (real part) of the nodes.

Quadrature Rule Data

Nodes

Node (ball center) Error (ball radius)
15.785897486802154794 - 4.2990374700625462625e-1014j 5.66e-490, 5.66e-490j
16.956688977518115821 - 2.2218383490856566844e-1020j 7.46e-492, 7.46e-492j
-16.346907708030889858 - 4.0286012922688726127e-1026j 8.5e-491, 8.5e-491j
14.283296852839775263 - 1.1285949787969575861e-1020j 3.59e-488, 3.59e-488j
14.760898438080646372 - 7.1896534592266283929e-1034j 1.17e-488, 1.17e-488j
17.641833959614263588 - 3.6713070878660378286e-1047j 4.24e-493, 4.24e-493j
-16.956688977518115821 + 1.3597711107148016678e-1054j 7.49e-492, 7.49e-492j
-18.472001959182435319 - 1.5932002275754868494e-1057j 8.51e-495, 8.51e-495j
-5.2790458545964047247 - 3.3949407458297914281e-1054j 2.66e-491, 2.66e-491j
3.122880221528317598 + 2.1856383419690724657e-1059j 2.01e-496, 2.01e-496j
16.346907708030889858 - 2.2638819606054340268e-1048j 7.32e-491, 7.32e-491j
18.472001959182435319 - 1.5190481495723170659e-1059j 8.14e-495, 8.14e-495j
15.259963931180608873 + 3.1562918679060839269e-1065j 2.79e-489, 2.79e-489j
-7.3376695957925152622 + 6.1269563679906215311e-1077j 2.51e-488, 2.51e-488j
-12.113350278147043781 - 2.9167646445029326914e-1074j 9.75e-487, 9.75e-487j
2.5263139009155984301 + 1.7372733135260821819e-1093j 1.41e-498, 1.41e-498j
-6.6936700909041694021 + 7.8247833919897349224e-1083j 7e-489, 7e-489j
-8.3652701559854143235 + 4.8159346804811294547e-1080j 1.6e-487, 1.6e-487j
-4.3385426828909078567 + 1.921924939887535727e-1086j 9.01e-493, 9.01e-493j
-9.0734247034499340129 + 6.4724734067745995533e-1080j 4.36e-487, 4.36e-487j
-10.545845158803879669 + 4.520766062922064838e-1088j 1.37e-486, 1.37e-486j
1.6365190424351079992 + 8.0931080173210649755e-1128j 1.23e-502, 1.23e-502j
10.927497868358537547 - 3.0633688453338812004e-1110j 1.43e-486, 1.43e-486j
8.3652701559854143235 + 1.0071439609520774991e-1127j 1.68e-487, 1.68e-487j
7.0083416616804912679 - 9.4682046387244610066e-1140j 1.26e-488, 1.26e-488j
-6.1611610480526666295 - 4.1387292016450309141e-1153j 1.69e-489, 1.69e-489j
-14.760898438080646372 + 1.5679036062150756465e-1149j 1.1e-488, 1.1e-488j
7.6749291229093645709 + 2.3544230237102786353e-1184j 4.73e-488, 4.73e-488j
11.710555666153284142 + 2.3901183656088798647e-1245j 1.25e-486, 1.25e-486j
-4.026350065134156794 - 5.0085556153360698103e-1288j 2.54e-493, 2.54e-493j
-6.4104762866297847371 + 3.0542858646211593607e-1281j 3.98e-489, 3.98e-489j
13.378314811672277855 - 9.5494636973720354092e-1294j 1.97e-487, 1.97e-487j
13.823365708473462062 + 2.1025598551205585084e-1318j 8.59e-488, 8.59e-488j
-13.823365708473462062 + 1.067166397846892189e-1335j 9.42e-488, 9.42e-488j
-9.7995903575560421661 - 2.2693386388036021625e-1336j 9.25e-487, 9.25e-487j
-3.7228517346656958725 + 1.5128495432714521639e-1342j 1.23e-494, 1.23e-494j
12.113350278147043781 - 2.0685965185628367332e-1333j 9.78e-487, 9.78e-487j
1.3467672487248763451 + 2.0584215967743989016e-1365j 7.54e-504, 7.54e-504j
-8.7171695182295643195 + 3.9216632453701249754e-1350j 2.73e-487, 2.73e-487j
-12.524807899786894676 + 4.4454917338807741349e-1350j 6.65e-487, 6.65e-487j
-1.6365190424351079992 - 1.2936453437713295468e-1364j 1.26e-502, 1.26e-502j
10.545845158803879669 - 4.2368078387430899314e-1348j 1.45e-486, 1.45e-486j
4.026350065134156794 - 3.3885598730483794869e-1367j 2.76e-493, 2.76e-493j
-4.9639840200927492835 - 5.5257964661369587896e-1366j 6.5e-492, 6.5e-492j
-15.785897486802154794 - 7.9593963637951838841e-1365j 5.56e-490, 5.56e-490j
-5.8918730094612552215 + 3.772530985705017203e-1365j 4.84e-490, 4.84e-490j
9.0734247034499340129 + 5.9037459112510718563e-1360j 4.3e-487, 4.3e-487j
7.3376695957925152622 - 2.469057730139058167e-1367j 2.53e-488, 2.53e-488j
1.0668273675044222989 - 7.3382902571294706807e-1395j 4.55e-505, 4.55e-505j
10.170027142471501039 + 2.0017981844360551201e-1375j 1.21e-486, 1.21e-486j
8.7171695182295643195 + 5.7368450350477889962e-1398j 2.68e-487, 2.68e-487j
-1.9315733801423747893 - 1.7126672971477471898e-1426j 2.16e-501, 2.16e-501j
-15.259963931180608873 + 7.9388549423902934687e-1411j 2.66e-489, 2.66e-489j
5.5914067726161139295 + 5.4862136747243302162e-1419j 1.27e-490, 1.27e-490j
11.315522849451234119 + 1.7524444550335827922e-1447j 1.46e-486, 1.46e-486j
-14.283296852839775263 + 3.764931932820004187e-1468j 3.29e-488, 3.29e-488j
-2.5263139009155984301 + 5.4325861828107272577e-1488j 1.51e-498, 1.51e-498j
12.946017675662147328 - 1.1160709160601162201e-1473j 4.03e-487, 4.03e-487j
3.7228517346656958725 - 5.5787359461472979501e-1492j 1.28e-494, 1.28e-494j
5.8918730094612552215 - 6.275431030623733679e-1488j 5.12e-490, 5.12e-490j
-5.5914067726161139295 + 4.4616212273069437034e-1490j 1.21e-490, 1.21e-490j
-17.641833959614263588 - 8.7780274653404577675e-1493j 4.16e-493, 4.16e-493j
-10.170027142471501039 + 7.942074240128988293e-1485j 1.18e-486, 1.18e-486j
-4.6497736192645573634 + 4.2733766200078987553e-1490j 1.88e-492, 1.88e-492j
4.3385426828909078567 - 1.834781175343575582e-1490j 8.87e-493, 8.87e-493j
-2.2288115845706893975 - 4.3969081511791051297e-1498j 5.03e-500, 5.03e-500j
4.1445471861258943321 + 1.8843836512138759657e-1490j 6.42e-493, 6.42e-493j
-2.8024858612875416991 - 9.3885600768106212393e-1495j 1.35e-496, 1.35e-496j
-13.378314811672277855 - 4.923444111596804386e-1485j 2.08e-487, 2.08e-487j
9.7995903575560421661 + 8.6567064191900315165e-1483j 8.86e-487, 8.86e-487j
2.8325455487991088469 + 1.0988339734223808454e-1499j 1.9e-496, 1.9e-496j
5.2790458545964047247 + 8.6862851931404203433e-1494j 2.87e-491, 2.87e-491j
-11.710555666153284142 - 5.1198374303126968307e-1490j 1.29e-486, 1.29e-486j
-4.1445471861258943321 - 4.9122537581128752129e-1496j 6.43e-493, 6.43e-493j
8.0177563925714241045 + 5.7156646937295699624e-1490j 9.25e-488, 9.25e-488j
0.53907981135137510807 - 4.2410072640512969642e-1510j 2.13e-507, 2.13e-507j
-10.927497868358537547 - 3.5034642913563300974e-1490j 1.44e-486, 1.44e-486j
-2.8325455487991088469 - 7.9010191955662142875e-1500j 1.82e-496, 1.82e-496j
-0.79968471368059122981 - 4.6431891225581650229e-1509j 3.33e-506, 3.33e-506j
-9.4341573461860495176 - 3.0011747212064521828e-1490j 6.83e-487, 6.83e-487j
-11.315522849451234119 - 6.8067671573128208398e-1490j 1.41e-486, 1.41e-486j
-7.0083416616804912679 - 3.7719864258997730972e-1492j 1.27e-488, 1.27e-488j
6.4104762866297847371 - 9.8577138174005781225e-1492j 3.86e-489, 3.86e-489j
-1.0668273675044222989 + 7.5164892634788038251e-1508j 4.26e-505, 4.26e-505j
9.4341573461860495176 + 2.3218909249500743086e-1488j 6.53e-487, 6.53e-487j
6.1611610480526666295 + 7.8031255877876086226e-1501j 1.77e-489, 1.77e-489j
-3.122880221528317598 + 4.7054252506217955914e-1508j 2.08e-496, 2.08e-496j
3.4215630331028556842 + 2.7863146943501045855e-1507j 1.31e-495, 1.31e-495j
-3.4215630331028556842 - 1.7783956364820171875e-1507j 1.41e-495, 1.41e-495j
-0.53907981135137510807 - 1.6419673542480980525e-1519j 2.39e-507, 2.39e-507j
2.8024858612875416991 - 4.9726512847353096009e-1508j 1.37e-496, 1.37e-496j
4.9639840200927492835 - 8.5269651196400259248e-1504j 6.84e-492, 6.84e-492j
0.27313019751791923672 - 4.5352154042074174322e-1520j 9.7e-509, 9.7e-509j
12.524807899786894676 - 2.3966442726408982828e-1497j 6.72e-487, 6.72e-487j
-0.27313019751791923672 - 2.3998676180178580287e-1526j 9.73e-509, 9.73e-509j
-5.7148951811103292396e-1586 - 8.7227380201337078171e-1586j 1.09e-1583, 1.09e-1583j
1.9315733801423747893 + 7.0681787514967524443e-1519j 2.18e-501, 2.18e-501j
0.79968471368059122981 + 8.3685370980873924545e-1524j 3.17e-506, 3.17e-506j
4.6497736192645573634 - 3.8475302404570156179e-1510j 1.95e-492, 1.95e-492j
-7.6749291229093645709 + 2.1267448626121932384e-1506j 4.7e-488, 4.7e-488j
-8.0177563925714241045 + 3.6442448330876465143e-1506j 9.01e-488, 9.01e-488j
-12.946017675662147328 + 1.5165735561058946062e-1504j 4.06e-487, 4.06e-487j
6.6936700909041694021 - 1.355611172317866789e-1506j 7.37e-489, 7.37e-489j
2.2288115845706893975 + 4.7069358794738455336e-1518j 4.5e-500, 4.5e-500j
-1.3467672487248763451 + 7.554597324101660046e-1522j 7.22e-504, 7.22e-504j

Weights

Weight (ball center) Error (ball radius)
1.6703942480753678412e-55 + 6.1554082996826794409e-1068j 2.49e-121, 7.06e-365j
9.369820803921567052e-64 - 2.1462195773098843043e-1073j 8.03e-125, 2.28e-368j
2.18656907562925146e-59 - 1.7521981599299359795e-1072j 8.02e-126, 2.28e-369j
9.345549724357493975e-46 - 9.6153079573596837969e-1064j 9.86e-118, 2.8e-361j
9.4630711737501745193e-49 + 3.4684136111436925318e-1065j 5.22e-119, 1.48e-362j
7.6869711523480357144e-69 + 2.5538282614791027313e-1076j 1.56e-127, 4.44e-371j
9.369820803921567052e-64 + 7.6427836149259025113e-1075j 2.01e-129, 5.71e-373j
3.0876737061625459398e-75 + 5.3977956486067462405e-1081j 1.01e-133, 2.87e-377j
1.1145404896343301095e-07 + 1.2401931782782684468e-1040j 8.69e-82, 2.47e-325j
0.00090538585591675386454 + 7.661955539027153611e-1037j 5.69e-60, 1.62e-303j
2.18656907562925146e-59 + 1.0088783396105574369e-1070j 2.85e-124, 8.1e-368j
3.0876737061625459398e-75 - 6.9030574014053442002e-1080j 2.09e-131, 5.94e-375j
5.5382141645869394059e-52 - 1.1326842931166687777e-1066j 1.96e-121, 5.57e-365j
2.7102765390315179259e-13 - 4.2091206783365576949e-1045j 6.03e-99, 1.71e-342j
2.227113208459728575e-33 + 5.2744660620990987265e-1058j 8.68e-120, 2.46e-363j
0.0048743640625120981992 - 1.9932847409018502183e-1036j 8.11e-58, 2.3e-301j
2.2521629318227708385e-11 - 2.0568394744399304075e-1043j 1.24e-95, 3.51e-339j
8.8954238269499697681e-17 + 1.0677293452031833077e-1047j 6.41e-105, 1.82e-348j
1.0063323216481304068e-05 - 1.8961891486051434897e-1038j 5.58e-81, 1.58e-324j
1.8978729542620040513e-19 + 1.656496902525742065e-1049j 1.4e-108, 3.97e-352j
1.0694334756940787119e-25 + 1.8896965598024118935e-1053j 3.93e-115, 1.12e-358j
0.030640148053354935869 + 1.9093058923704846262e-1036j 2.85e-48, 8.08e-292j
1.8050186519902053956e-27 - 8.7746623122825083922e-1054j 4.79e-118, 1.36e-361j
8.8954238269499697681e-17 + 3.4706207598490024209e-1047j 4.17e-109, 1.18e-352j
2.7896307605315810826e-12 + 7.7618705235973608805e-1044j 3.13e-103, 8.88e-347j
5.708599015201947904e-10 - 4.0452925335221146501e-1042j 6.44e-97, 1.83e-340j
9.4630711737501745193e-49 + 1.1391795753976575722e-1066j 7.48e-132, 2.12e-375j
2.1967045727854614585e-14 + 1.695386424439507394e-1045j 1.81e-106, 5.13e-350j
2.6466354916011764545e-31 - 5.7381632920724489312e-1056j 2.58e-121, 7.32e-365j
3.6286587123426308491e-05 - 9.7852661787697867196e-1038j 1.56e-85, 4.41e-329j
1.2424878956131412985e-10 + 1.1413455008100564173e-1042j 6.13e-99, 1.74e-342j
2.3871592921513496729e-40 - 6.5951414530455697463e-1061j 9.11e-126, 2.59e-369j
5.7870702713546416648e-43 + 2.7571579720545142432e-1062j 2.19e-126, 6.23e-370j
5.7870702713546416648e-43 + 1.8680900311040007966e-1063j 1.07e-131, 3.03e-375j
2.0584315405258572041e-22 + 2.0441878444968497503e-1051j 1.05e-117, 2.98e-361j
0.00011815611907325418817 + 8.7469237189553954483e-1038j 9.07e-88, 2.57e-331j
2.227113208459728575e-33 + 3.9896246932822267111e-1057j 2.06e-124, 5.84e-368j
0.046016643329317246699 - 2.4796958106150518406e-1036j 6.35e-61, 1.8e-304j
4.4584556676147601414e-18 - 1.3630959637462057418e-1048j 6.32e-113, 1.79e-356j
1.4324485205099824803e-35 - 2.8722838037136124029e-1059j 8.06e-128, 2.29e-371j
0.030640148053354935869 + 1.5508731868550747264e-1036j 3.74e-69, 1.06e-312j
1.0694334756940787119e-25 + 9.4739545201419206331e-1053j 2.22e-121, 6.31e-365j
3.6286587123426308491e-05 - 1.6478830231672259076e-1037j 1.44e-95, 4.08e-339j
5.603924585054046255e-07 - 4.9808602673715438002e-1040j 2.46e-97, 6.99e-341j
1.6703942480753678412e-55 + 2.258481960311108636e-1070j 4.04e-138, 1.15e-381j
3.3339578355604581906e-09 + 1.1635619498003385886e-1041j 1.65e-101, 4.7e-345j
1.8978729542620040513e-19 + 6.1254552507696238735e-1049j 1.46e-118, 4.13e-362j
2.7102765390315179259e-13 - 1.1509298178380227057e-1044j 6.09e-113, 1.73e-356j
0.061771494980788923077 + 3.3000589237953180962e-1036j 5.71e-67, 1.62e-310j
5.1673354875050576607e-24 - 9.4333178433423782629e-1052j 1.01e-121, 2.86e-365j
4.4584556676147601414e-18 - 4.7184989539696225055e-1048j 1.87e-117, 5.3e-361j
0.018316002636362980945 - 1.2435956525379283282e-1036j 2.87e-83, 8.13e-327j
5.5382141645869394059e-52 - 1.8984134119097856144e-1068j 2.33e-139, 6.62e-383j
2.0030954118317240368e-08 - 7.4318642760921726782e-1041j 8.03e-110, 2.28e-353j
2.4534882703315410435e-29 + 7.4425048117347049322e-1055j 9.99e-125, 2.84e-368j
9.345549724357493975e-46 - 5.1842818676618235295e-1065j 4.82e-137, 1.37e-380j
0.0048743640625120981992 - 1.4436813933355592891e-1036j 9.25e-93, 2.63e-336j
6.8721777377996608355e-38 + 1.3656562903388252695e-1059j 1.11e-129, 3.16e-373j
0.00011815611907325418817 + 1.4140176561285179507e-1037j 3.34e-102, 9.49e-346j
3.3339578355604581906e-09 + 2.5475555926726093548e-1041j 1.18e-111, 3.34e-355j
2.0030954118317240368e-08 - 3.5464702997355883521e-1041j 3.35e-108, 9.5e-352j
7.6869711523480357144e-69 - 1.413179789627788156e-1077j 2e-148, 5.68e-392j
5.1673354875050576607e-24 - 2.0452145649673291299e-1052j 4.26e-126, 1.21e-369j
2.522231645745126302e-06 + 2.4152362945201861052e-1039j 1.37e-105, 3.89e-349j
1.0063323216481304068e-05 - 3.331910696675476187e-1038j 6.18e-108, 1.75e-351j
0.0099055320039445256783 + 1.144933791002589351e-1036j 6.39e-96, 1.82e-339j
6.7846775685399321005e-07 + 1.3782802241360858465e-1037j 1.17e-106, 3.31e-350j
0.00063248620154442368977 + 8.8174509829849107295e-1036j 8.9e-100, 2.53e-343j
2.3871592921513496729e-40 - 5.504276254571200758e-1062j 3.64e-136, 1.03e-379j
2.0584315405258572041e-22 + 8.7207394120588574595e-1051j 1.36e-124, 3.85e-368j
0.0015820512578100746484 - 1.1986957499649029661e-1035j 9.75e-103, 2.77e-346j
1.1145404896343301095e-07 + 2.4849191127902314017e-1040j 7.32e-113, 2.08e-356j
2.6466354916011764545e-31 - 8.5351156108676520642e-1057j 2.16e-132, 6.13e-376j
6.7846775685399321005e-07 + 8.0541528097128807382e-1038j 1.75e-107, 4.97e-351j
1.5130881564371224978e-15 - 2.4589238489689693907e-1046j 2.38e-121, 6.77e-365j
0.090315781292036387651 + 4.679886817049249098e-1036j 1.5e-95, 4.26e-339j
1.8050186519902053956e-27 - 1.6000615773015840375e-1054j 1.49e-130, 4.23e-374j
0.0015820512578100746484 - 8.3421010150086408093e-1036j 2.93e-105, 8.32e-349j
0.075863060466035555997 - 3.7619227677919799173e-1036j 4.95e-99, 1.4e-342j
6.823959137466846397e-21 - 1.9008389059818569973e-1050j 1.75e-126, 4.98e-370j
2.4534882703315410435e-29 + 1.231369268952828842e-1055j 9.07e-132, 2.58e-375j
2.7896307605315810826e-12 + 2.9916767884886071706e-1044j 6.48e-120, 1.84e-363j
1.2424878956131412985e-10 + 2.699981085783423021e-1042j 6.88e-122, 1.95e-365j
0.061771494980788923077 + 2.8825602353656917753e-1036j 1.08e-105, 3.05e-349j
6.823959137466846397e-21 - 7.5348322617372520576e-1050j 8.89e-129, 2.52e-372j
5.708599015201947904e-10 - 9.2174245503650739564e-1042j 3.78e-121, 1.07e-364j
0.00090538585591675386454 + 5.1328123181436826334e-1037j 3.07e-112, 8.72e-356j
0.00034329382297285874106 - 2.6803641954153483053e-1037j 3.18e-115, 9.03e-359j
0.00034329382297285874106 - 1.7260436830830072956e-1037j 3.49e-113, 9.89e-357j
0.090315781292036387651 + 4.3710456908726929837e-1036j 2.25e-109, 6.37e-353j
0.00063248620154442368977 + 1.2620298826312477278e-1035j 3.09e-113, 8.76e-357j
5.603924585054046255e-07 - 9.5449010456087776006e-1040j 1.3e-119, 3.7e-363j
0.10389130241895943778 - 4.7557574762223930944e-1036j 6.86e-111, 1.95e-354j
1.4324485205099824803e-35 - 2.4800631809720429908e-1058j 4.04e-139, 1.15e-382j
0.10389130241895943778 - 4.5941118128572654904e-1036j 1.98e-111, 5.63e-355j
0.10954810191498812194 + 4.67006216517931126e-1036j 2.31e-111, 6.54e-355j
0.018316002636362980945 - 1.5900485416544893203e-1036j 9.56e-115, 2.71e-358j
0.075863060466035555997 - 4.1630724272957912427e-1036j 1.98e-112, 5.62e-356j
2.522231645745126302e-06 + 4.4298524167323347486e-1039j 7.22e-119, 2.05e-362j
2.1967045727854614585e-14 + 5.8676670294783120438e-1046j 1.19e-127, 3.38e-371j
1.5130881564371224978e-15 - 8.0338850605594890027e-1047j 1.27e-128, 3.62e-372j
6.8721777377996608355e-38 + 1.359652125326984539e-1060j 1.17e-142, 3.32e-386j
2.2521629318227708385e-11 - 5.0809694582372148214e-1043j 1.49e-125, 4.22e-369j
0.0099055320039445256783 + 1.5210460414982296659e-1036j 1.78e-118, 5.25e-362j
0.046016643329317246699 - 2.0901303143195237805e-1036j 3.26e-118, 7.76e-362j

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All rules of Hermite (probabilists') type can be downloaded here packed into a single file.